Complex numbers have their origin in the solution of equations, for which the real number system proved inadequate. The fundamental theorem of algebra emphasises the importance of extending the real number system to the larger complex number system - "every polynomial equation of degree n with complex coefficients has n roots in the complex numbers".

Complex numbers are considered in cartesian (rectangular) and polar form. In polar form, multiplication and division of complex numbers are easily represented. The concept underlying powers is also straightforward (de Moivre's theorem) and provides a method of evaluating roots of complex numbers.

A consideration of the quadratic recurrence relation z_n=z_n-1²+c leads to the definition of the Mandelbrot set and the study of chaos theory.

notes - introduction

notes - polar form and de Moivre's theorem

resources page - complex numbers

resources page - Mandelbrot and Julia sets